# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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### Confusion regarding vector/matrix multiplication in index notation

I came across this question and answer (sorry I don't have an electronic source for it, only a paper copy). After reading the answer it had me questioning the notation one uses to denote row/column ...
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### $\det(A^2-B^2) \leq \det(A^2)$ when $B$ is of rank 1

This is an exercise problem given at linear algebra class. As a problem before this I was able to show that $$\det(A + uv^T) = \det A + v^T (adj A )u$$ when $A$ is order $n$ square matrix and $u,v$ ...
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1 vote
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### Reading $B^{-1}$ from simplex table

In uni I'm following a course on optimalisation and I have come across a problem. I am given the following minimalisation problem: and the corresponding final Simplex table: I now need to determine ...
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### 3x3 real matrix decomposition to SVD using two unit quaternions and scale vector

I've been trying to search about doing 3x3 real matrix SVD, but instead of decomposing it into matrices, represent the two rotations as unit quaternions with the singular values as separate scale ...
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### Further maths matrix question [closed]

The matrices p = [3/7, 3/7][1/7, 3/7] and Q = [11, 3][5,2] and R = [0,0][1,1] represent the transformations T, U and V respectively. a. A single transformation W is obtained by combining these ...
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### How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Known $(B,C_0,C_1)$, how to calculate ...
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1 vote
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### How to go about finding the minimal conditions guaranteeing that such a matrix is invertible?

I am working on an economic input-output model, and I want to find the conditions under which a system of linear equations determining the equilibrium yields a unique solution. I have an equation of ...
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### Further explanation wanted on 'double Gaussian elimination' to triangularize a matrix.

I am trying to learn a more efficient way to triangularize a matrix. I found the following answer here on StackExchange which I found interesting, talking about 'double Gaussian elimination': Short ...
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### $n$ linear equations for $n+1$ unknowns

Let us consider a linear system \begin{eqnarray*} a_{11}x_1 + \dots + a_{1n}x_n + b_1y &=& c_1 \\ &\dots&\\ a_{n1}x_1 + \dots + a_{nn}x_n + b_ny &=& c_n \end{eqnarray*} for ...
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### Using adjacency matrices to calculate graph rotations [closed]

Given the adjacency matrix $A$ of a tree $T$, is there a way of transforming that adjacency matrix in an efficient way to perform a rotation on that graph?
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### If $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup\mathbb{T}$, does $\text{dim ker}(AB-BA)=\text{dim ker}(A^*B-BA^*)$?

I already asked this question for general $B$ and it was answered negatively here, so if the statement is true, one has to use the assumption on $B$ as well. In the original question I already ...
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### Does square root of matrices obey matrix inequalities? [closed]

Consider two positive semi-definite matrices $A,B$ such that $A\geq B \geq 0$. Then, is it true that: $A^{1/2} \geq B^{1/2}$?
1 vote
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### Uniqueness of Hessenberg matrix in Cholesky factorization of Hankel matrix

I'm reading the paper how bad are Hankel matrices?, where the author claimed the following: Lemma 2.1. For any positive definite Hankel matrix $H \in \mathbb{R}^{n \times n}$ there exist a ...
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1 vote
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### Proving a matrix is diagonalizable

Consider the matrix $M\in M_3(\mathbb{R})$ $$M=\begin{pmatrix} 5 & -6 & -6 \\ -1 & 4 & 2\\ 3 &-6&-4 \end{pmatrix}.$$ I need to show that $M$ is diagonalizable, and find a ...
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### Transform a column vector into its transpose with matrix multiplication? [closed]

I don't have tensor in mind here, what follows is just a question about linear algebra: If I start out with an $n\times 1$ column vector over the reals, are there any combinations of matrix ...
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### If a singular square matrix is multiplied by a non-singular square matrix, the null space of the result is what?

If a non-singular square matrix $B$ is multiplied by a singular square matrix $A$ of the same order, the nullspace of the resulting matrix $C=B\times A$ or $C'=A \times B$ remains unchanged from that ...
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